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Download the Six Sigma Handbook Third Edition

admin — Mon, 23 Nov 2009 11:46:45 +0000

The authoritative classic–revised and updated for today’s Six Sigma practitioners

Whether you want to further your Six Sigma training to achieve a Black or Green Belt or you are totally new to the quality-management strategy, you need reliable guidance. The Six Sigma Handbook, ThirdEdition shows you, step by step, how to integrate this profitable approach into your company’s culture.

Co-written by an award-winning contributor to the practice of quality management and a successful Six Sigma trainer, this hands-on guide features:
- Cutting-edge, Lean Six Sigma concepts integrated throughout
- Completely revised material focused on project objectives
- Updated and expanded problem-solving examples using Excel and Minitab
- A streamlined format that puts proven practices at your fingertips

The Six Sigma Handbook, Third Edition is the only comprehensive reference you need to make Six Sigma work for your company. The book explains how to organize for Six Sigma, how to use customer requirements to drive strategy and operations, how to carry outsuccessful project management, and more. Learn all the management responsibilities and actions necessary for a successful deployment, as well as how to:
- Dramatically improve products and processes using DMAIC and DMADV
- Use Design for Six Sigma to create innovative products and processes
- Incorporate lean, problem-solving, and statistical techniques within the Six Sigma methodology
- Avoid common pitfalls during implementation

Six Sigma has evolved with the changing global economy, and The Six Sigma Handbook, Third Edition is your key to ensuring that your company realizes significant gains in quality, productivity, and sales in today’s business climate.

560 pages

Six Sigma Handbook Third Edition

McGraw-Hill Professional;
3 edition (September 21, 2009)
ISBN: 0071623388
PDF 9.6Mb

Download it from http://rapidshare.com/files/310019896/McGraw-Hill_-_The_Six_Sigma_Handbook_3rd_Edition__2009_.zip or

http://www.10xdownloads.com/servers.asp?pb=2&PID=2bc60e94-1b70-4fdc-9f6f-d525c70aaf4f&jstyle=3&q=The%20Six%20Sigma%20Handbook,%203rd%20Edition

Six Sigma and Minitab A complete Toolbox Guide for All Six Sigma Practitioners

admin — Wed, 11 Nov 2009 11:29:08 +0000

Six Sigma and Minitab

Six Sigma and Minitab: A complete toolbox guide for all Six Sigma practitioners (2nd edition) by Quentin Brook

In completing my final week of black belt training, my esteemed master black belt, George Woodley. Thanks George.

Description

The long awaited 2nd edition of this bestselling pocket guide on Six Sigma contains a range of new tools and techniques including new Lean material and Improve tools. In addition it has been updated for Minitab 15 (whilst still compatible with versions 13 and 14). Cutting through Six Sigma s strange terminology and consultancy speak, this guide delivers Six Sigma in a down to earth, logical and user friendly format. Minitab: For each tool, this guide details how to enter the data into Minitab, interpret the results and avoid the common pitfalls. Interactive: All the data files and templates are available online. Routemaps: A logical flow is provided through each DMAIC phase. Who should use this book? Six Sigma Trainees: Both during and after training, this guide provides an invaluable reference text to those who are actually implementing Lean Six Sigma improvement projects. Six Sigma Project Sponsors and Managers: For those who are accountable for deploying Six Sigma or sponsoring Six Sigma projects, and who might not have been fully Six Sigma trained, this guide will provide an overview of the tools and techniques that your project teams are trained in.

Here are some customer reviews we spotted on Amazon.

I really like how this “toolbox” book is put together. It is quick and to the point in a compact format. It shows what minitab tests can be used within each phase of the DMAIC process. I recommend it for the person who already has at least a bit of an understanding of the Six Sigma philosophy. It is the kind of book you will use for minitab reference over and over again.

This book is my go-to resource for DMAIC processes. It provides simple explanations of the tools and outcomes and is the reference material I always recommend to new belts.

Really good book for anybody using Minitab in Six Sigma or just plain statistics. Not too superficial (like many ‘Management Books’) and not too many silly examples (like some ‘Text Books’). Really useful. After reviewing it we bought one copy for each Black Belt at our company.

I do agree, however, that the new listed price is $49.99 given that much of Minitab’s documentation is now complete and freely available on their website.

Spiral-bound: 240 pages

Publisher: QSB Consulting (October 23, 2006)

Language: English

ISBN-10: 0954681320

ISBN-13: 978-0954681326

Also, if you’re looking for a great Master Black Belt, mentor, statistician, I highly recommend George Woodley. George learned Six Sigma at Motorola, taught SPC at Ford but to name a few of his stints. Check out his website at http://qualityforall.net

Introduction to Six Sigma Statistics

admin — Mon, 19 Oct 2009 17:04:26 +0000

I thought I’d bring in some key points here before embarking on a serious Design of Experiment, or multiple regression analysis. I’ve come across several black belt candidates in training (like myself), who get too caught up with the statistical side of things (or Minitab) and forget the basics. I thought

1. Learn the scales of measure

Nominal
It is in the name
Marital status, Phone numbers

Ordinal
Relative, unequal value ranking
Race finish, opinion poll response

Interval
Equal intervals are equal differences
Calendar year, Fahrenheit temperature

Ratio
Proportional amount of difference
Has a real zero value
Annual income, Kelvin temperature

2. Learn the measures of central tendency

Mean: Arithmetic average of a set of values
• Reflects the influence of all values
• Strongly Influenced by extreme values
Mode: The most frequently occurring value
Median: Midpoint in a string of sorted data, where 50% of the observations, or values, are below and 50% are above
• Does not necessarily include all values in calculation
• Is “robust” to extreme scores
• Organize the data from low values to high when determining the Median

3. Learn the measures of dispersion

Range: the distance between the extreme values of a data set (Highest – Lowest)
The range is more sensitive to outliers than the variance

Variance: the Average Squared Deviation of each data point from the Mean

Standard Deviation: the Square Root of the Variance
measure of the average deviation about the mean

4. Understand the different types of data

Variable (quantative) and Continuous Data (Decimal subdivisions are meaningful)
o Time (seconds)
o Pressure (psi)
o Conveyor Speed (ft/min)
o Rate (inches

Attribute (qualitative)
o Categories
o Good/Bad (Pass/Fail)
o Machine 1, Machine 2, Machine 3
o Shift number
o Counted things (# of Errors in a document, # units shipped, etc.)

Convert Attribute to Continuous wherever possible. Here are some examples of attribute data converted to continuous data:

– Count of defects to ‘% defects’
– Y/N late to ‘average days late’
– Leaks/No leaks to ‘rate of leaks on a continuous scale’
– Success or failure of electrical parts to ‘voltage flow of good parts’

5. Descriptive Statistics

Consists of basic statistics and graphical techniques used to summarize data
Measures of central location
Measures of spread (dispersion)
Evaluation of symmetry & skewness
Typical graphical techniques
- Histograms
- Boxplots
- Dotplots
- Normal probability plots

6. Be able to identify different data distributions

Why does it matter? If you’re treating non-normal data as normal, you’ll get a totally different p-value for your data set, thereby overstating or understating your prediction or analysis.

Normal Distribution (Bell Curve)

The “Normal” Distribution is a distribution of data which has certain consistent properties (the mean, median and mode are equal in value)

These properties are very useful in our understanding of the characteristics of the underlying process from which the data were obtained

Most natural phenomena and man-made processes are distributed normally, or can be represented as normally distributed

The Normal Distribution is a continuous distribution which is symmetrical and extreme values are less likely than moderate values (unimodal)

An example would be measuring heights of people or the length of a table. In either case the measurement is continuous and can be broken down into finer increments

t-distribution

The t distribution assumes samples are drawn from a normal distribution but the population variance, s2, is not known… The shape of the t-distribution varies as the sample size, n, changes. The distribution becomes more narrow as the sample size becomes larger. As n becomes very large, the critical value corresponding to the area under the curve approaches the Normal distribution’s Z value

Poisson Distribution

Appropriate as a model of number of defects or nonconformities in a unit of product
X is number of defects found in a per unit basis
– Per unit area, per unit volume, per unit time, etc.
– Area X is a discrete, positive integer
– Area for opportunity is a finite region of space, time or product

When the average is high, the distribution can be approximated by the normal distribution

When the average is low, the distribution is skewed to the right

F Distribution

A continuous distribution formed from the ratio of variances calculated from two independent samples drawn from Normal Distributions

Chi-square Distribution

A continuous distribution used in statistical hypothesis testing and confidence interval estimation for many different applications, including inferences about a population variance

7. Know about samples and population sizes

Population is every possible observation (census)

Samples are subsets of populations

Data is obtained using samples because we seldom know the entire population

Descriptive statistics apply to any distribution
- Sample or population

Population statistics are desired, but often not available

Samples from a population can be used to ‘infer’ or approximate population parameters

8. Know the Statistics and Reporting Tools

By far the most used Six Sigma tools is Microsoft Excel. Excel provides most of the day to day uses required to manage most Six Sigma projects. Please note though that Excel must be accompanied by other Microsoft tools such as Word, PowerPoint, and Outlook for email. In other words, the Six Sigma Black or Green belt candidate should always ensure that he or she has the latest version of Microsoft Office installed.

Also, check with your IT department or on the Microsoft Excel disc for the “Data Analysis Pack” that will allow you, through Excel to perform:

– Anova
– Correlations
– Covariances
– Descriptive Statistics
– Exponential Smoothing
– F-Tests and Two-Samples for Variances
– Fourier Analysis
– Histograms
– Various t-tests

However, the ultimate tool for Six Sigma Black Belts is MiniTab. Minitab will get you going where Excel leaves you hanging.

– Minitab will go into profound levels of englightenment with
– Measurement System Analysis
– Multivariate Analysis
– Anova
– Regression Analysis
– Statistical Process Control
– Reliability/Survival Analysis
– And other great simulations

Don’t forget to check out Minitab’s powerful reporting tools that will provide the following graphical reports:

– Dotplots / Histograms / Normal Plots
– Run charts / Time Series
– Pareto Diagrams
– Stratification (2nd Level Pareto)
– Boxplots
– Scatter Plots
– Checksheets / Concentration Diagrams

Now you should be ready to go run your multi-var charts, multiple regression, 2k full-factorial regression, and last but not least, your Design of Experiment. We’ll keep data transformations to another paper.

Please don’t hesitate to drop us a line at admin@sixsigmaz.com if you have any questions or concerns.

Introduction to Design of Experiments

admin — Mon, 21 Sep 2009 16:54:23 +0000

The Basic Concepts Underlying DOE

Six Sigma stresses making alterations in existing business processes for improving overall efficiency. To make effective alterations, it is important to first understand the various aspects of business processes so that the cause and effect relationship between various processes can be determined. However, this is easier said than done because most business processes comprise of multiple sub processes, which themselves are quite complicated.

Making alterations in a simple business process having just two to three sub-processes is easy, but the task becomes quite difficult when alterations are to be made in a business process having more than ten sub-processes. The level of difficulty goes on increasing with an increase in the number of sub-processes. This is where DOE is beneficial as it can easily crawl through vast amounts of data that is generated as part of the six-sigma implementation process. This data is meant for understanding the cause and effect relationship between various processes and sub-processes, but since the amount of data generated is quite huge, advanced tools such as DOE are used.

Why Is DOE Necessary?

DOE analyses the data and generates quantifiable results, which are then used for defining the type of experiments or alterations that are to be conducted for achieving Six Sigma quality levels. Conducting pre-tests or experiments is necessary in Six Sigma because the organizational efficiency and productivity are at stake while the implementations are going on. If Six Sigma concepts and methodologies are not implemented properly, it can seriously affect the company’s bottom-line and lead to redundancies.

Another thing is that organizations cannot hope to achieve desired results just by conducting wayward and misguided experiments or alterations. DOE is effective because it helps Six Sigma professionals in selecting the most suitable experiment designs that in turn would help in achieving the desired results. In the absence of DOE, it would be quite difficult to ascertain the type of experiments that are to be conducted.

Other Uses And Benefits

DOE can also be used for understanding the root cause of variations in a business process. Business processes are designed for delivering exactly the same quality and quantity, irrespective of what they are being used for, be it for manufacturing a product or rendering a service. However, maintaining this level of consistency is not always possible because the efficiency of business processes can be affected by various variable factors that are quite difficult to decipher.

This is where DOE can help because it can analyze vast amounts of data related to the variations and generate results that can be used for pointing out the root cause of variations. Once the variations have been eliminated, it would become quite easy for Six Sigma professionals to successfully carry out the implementations. DOE is thus an inseparable part of Six Sigma implementations in any type of industry.

Tony Jacowski is a quality analyst for The MBA Journal. Aveta Solution’s Six Sigma Online offers online six sigma training and certification classes for lean six sigma, black belts, green belts, and yellow belts.

Six Sigma and Baseball

admin — Thu, 27 Aug 2009 02:01:13 +0000

Six Sigma Baseball

With a much anticipated and needed vacation coming to an end I thought I would deliver on my personal promise and write a short article in order to better organize my 6s thoughts, which have been nothing less than tumultuous lately.

I’ve been troubled for a few months now regarding the future of Six Sigma and what direction my career should (could) take. If Motorola introduced what seemed to be the next best thing since sliced bread (baked, packaged, and delivered a la ISO9001 of course) and TQM, why hasn’t widespread adoption outside the business world occurred?

If Six Sigma is good enough for Motorola, GE, and Tyco and thousands of others, why isn’t it good enough for our governments, schools, and hospitals?

Malcolm Baldrige compensates for the lack of 6s in the healthcare industry but where are they with statistical analysis and SPC?

During my holidays I read a good book which seems to have piqued my boss’s interest for the shear Billy Beane Factor: Moneyball: The Art of Winning an Unfair Game by Michael Lewis, ISBN-13: 978-0393324815, published by W.W. Norton & Co.

Moneyball goes to great lengths to explain the Oakland A’s success at seasonal ball game wins on the lowest payroll in Big League ball.

As it turns out, the problem in professional baseball was that everyone was looking at the wrong stats. Let me take that back. More relevant stats needed to be made available and scouts had to stop recruiting on emotion and gut feeling and identify true talent. Up until then it didn’t seem that any statistical analysis had been extensively correlated. This screams measurement system analysis (MSA) to me.

Bill James realized this in the late seventies and his annual Baseball Abstract provided the Oakland A’s the necessary data and approach to the first non-official Six Sigma Baseball project.

Some of you are probably wondering where I’m going with this, or have read Moneyball (no mention of Six Sigma or SPC for that matter) but hold on:

Isn’t Six Sigma just another means to deliver the best possible perhaps even perfect? product, process, or service as possible at the lowest cost possible as quickly as possible? It’s as simple as y=f(x). Don’t bore me with Voice of the Customer (VOC), Quality Function Deployments (QFD) and Houses of Quality, Kanos, Designs of Experiments (DOE), Analysis of Variances (ANOVA), etc. I don’t want to oversimplify and denigrate the Six Sigma approach, however, let’s not complicate things either.

Here’s my DMAIC take on baseball.

Define

What are the Oakland A’s trying to accomplish? World Series? More ticket sales? Increased revenue from merchandising? Happier fans translates into increased ticket sales.

The catch here is that if fans want to see slugging and home runs, this doesn’t necessarily translate into wins. Read Moneyball to find out how fans react when established sluggers were told to walk bases.

Measure:

What are we measuring? Errors, On-Base Percentages? Runs? Strike-outs?

Analyze

What are the correlations that we need to identify to better analyze the game and increase wins. What works now may not work in the playoffs.

Understand that a batting average of .288 in the season, may actually be .201 when a certain batter faces a certain pitcher.

Do change-up pitches always work with the same batters? Does the speed of a fastball correlate to the number of strikeouts?

Improve

How do we make better trades and optimize the drafting process? Will a first-round draft pick be necessarily better than a sixteenth-round pick?

Control

Remember the Upper and Lower Control Limits Specifications for whatever it is you’re monitoring. The LCL and UCL can apply to the strike zone, the acceptable number of men on base with two outs, the batter’s count, etc.

Consider a pinch-hitter or a relief pitcher.

Conclusion

While much work has been done over the past twenty years with regards to the Six Sigma in improving manufacturing, product design, and service industries, with increasingly more resources being made available with software engineering, healthcare, and financial sectors, where are we with 6s in our personal lives? Wouldn’t it be great if we had our own personal toolboxes our neighbors used

Would anyone be so kind as to demonstrate the Cpk or sigma level of a baseball team, recruiting process, or any other interestingly fun spin on Six Sigma and baseball? Many Six Sigma practitioners are responsible for increasing efficiencies and reducing costs by hundreds of percentage points. Has anyone begun simple projects to monitor their fuel economy on their way to work, or find more efficient ways of saving time on their daily subway trip to work?

Six Sigma Glossary & Definitions

admin — Wed, 26 Aug 2009 11:06:56 +0000

Alpha Risk – Risk of concluding that two characteristics are different when they are actually the same.

Alternative Hypothesis – Statement of change or difference. The statement is assumed correct if the null hypothesis is not supported by the data.

ANOVA – Analysis of Variance – Hypothesis test used to determine whether two or more sample means are significantly different.

Attribute Data – Sometimes used in place of the term discrete data. Attribute data is qualitative data that has two outcomes. Examples include pass/fail, tastes good/tastes bad and acceptable/not acceptable. Attribute data is used specifically in reference to Attribute Control Charts, which are limited to counting and plotting defects or defectives.

Attribute Gage R&R – A Gage Repeatability & Reproducibility (Gage R&R) study used to determine the measurement variability when attribute data is used.

Backbone Test – A main supporting tool or factor used in statistical analysis.

Bartlett‘s Test – Type of hypothesis test that compares variances of two or more normally distributed populations.

Baseline – The “as-is” level of performance of the process being investigated.

Baseline Control Chart – A graph of the initial process performance over time that is used to detect whether common or special cause variation exists in a process.

Beta Risk – Risk of concluding that two characteristics are the same when they are actually different.

Between Variance – Between variance is based on the distances of the sample means from the overall mean. If the samples come from populations with equal means, one would not expect an undue amount of variation in the sample means.

Bias – Bias within a measurement system describes the difference between the observed average and a reference value of the measurements. Bias is related to accuracy.

Binomial Distribution – A distribution of probabilities used when there are only two independent outcomes to the hypothesis test (e.g. heads/tails, yes/no).

Box Plot – A type of plot which identifies variability and centering based on quartiles. It uses a rectangular box to represent the middle 50% of the data and whiskers to show the extent of the data. A Box Plot requires data from a continuous output and a discrete (typically attribute) input at one or more levels.

C Chart – A graphical display of number (counts) of defects in a subgroup (part or unit) with a constant sample size. A C Chart can be used to detect special cause variation.

Capability Analysis – A statistical measure which helps determine how well a process meets customers’ standards.

Capability Indices – Indices describing the overall effectiveness of a process in meeting specific criteria in both the short and long term.

C&E Analysis – Cause and Effect Analysis – The Cause and Effect Analysis (C&E Analysis) is an analysis in which either causes to an effect are brainstormed (the Cause and Effect Diagram), or multiple causes, which have already been identified, are prioritized against several criteria (the Cause and Effect Matrix). C&E Analyses are typically used when limited data is available

C&E Diagram – Cause and Effect Diagram – The Cause and Effect Diagram (C&E Diagram) is a fishbone-shaped diagram that helps a team brainstorm potential root causes of a defect. The problem to be solved is placed at the head of the fish and the six thought-generating categories – Personnel, Machines, Materials, Methods, Measurements and Environment – are placed at the ends of the fish bones.

C&E Matrix – Cause and Effect Matrix – The Cause and Effect Matrix (C&E Matrix) is a tool that allows one to prioritize many items when there are many criteria they need to be prioritized against.

Chi-Square Distribution – A sampling distribution used to determine the confidence interval for standard deviation and to perform Chi-Square Hypothesis Tests: Goodness of Fit Test and Test for Association.

Chi-Square Hypothesis Test – Hypothesis test used to test the goodness-of-fit between a sample and a hypothesized distribution, or the association of two or more variables.

Classical Yield – The ratio of the number of units that ultimately pass through the entire process to the number of units that enter into the process. See also Final Yield, which equals output divided by input.

Clustering – A type of special cause variation within a Run Chart in which the data set exhibits fewer runs around the median than expected, resulting in a “clustering” of data points in just a small range of values or a shift in the average values.

Coefficient of Determination – A measure of the correlation between the dependent and independent variables in a Regression Analysis.

Common Cause Variation – Process variability that is free of assignable cause. It is typically associated with short-term variability or subgroup variability. Also referred to as white noise or expected variation.

Confidence Interval – A range of numbers in which population parameters are likely to fall.

Confidence Level – The fixed probability of correctly accepting the null hypothesis.

Contingency Table – A table of observed frequencies used in Chi-Square Hypothesis Tests.

Continuous Data – Continuous data, sometimes referred to as variable data, is data that is measured on a continuum or a scale that can be meaningfully divided into finer and finer increments of precision. For example, length and weight can be measured to any desired level of precision.

Continuous Gage R&R – A Gage Repeatability & Reproducibility (Gage R&R) study used to determine the measurement variability when continuous data is used.

Control Chart – A process control tool, in which data is plotted and statistically analyzed in order to discern whether the process exhibits common and/or special cause variation.

Control Limit – Control Limits are typically set +/-3 standard deviations from the centerline to determine whether or not the process is in control.

Control Plan – A single document or set of documents that provides a point of reference among the Key Process Input Variables (KPIV’s), Key Process Output Variables (KPOV’s), specifications and instructions for the completed project. It documents the actions, including schedules and responsibilities that are needed to control the KPIV’s at their optimal settings.

Controllable Input – A process input that has adjustable settings, which can be modified by the project team during a project.

Correlated – Data identified as having a causal, complementary, parallel or reciprocal relationship.

Correlation – Statistical analysis that determines whether or not one variable can be used to predict another. Correlation does not necessarily prove causation.

Correlation Coefficient – A measure of the interdependence of two random variables that range in value from -1 to +1, indicating perfect negative correlation at -1, absence of correlation at zero and perfect positive correlation at +1.

Cp – A capability index that measures the potential capability of a process to meet expected specification limits, or tolerance levels, in the short term, assuming the process is ideally centered (regardless of where the process is actually centered).

Cpk – A capability index that measures the potential capability of a current process to meet expected specification limits, or tolerance levels, in the short term, using the current process average.

Critical Input Variable – An input variable (X) that has been statistically proven to impact the process output (Y).

CTC – Critical to Cost – Reference to a product, service and/or transactional characteristic that significantly influences a customer in terms of cost.

CTD – Critical to Delivery – Reference to a product, service and/or transactional characteristic that significantly influences a customer in terms of delivery.

CTQ – Critical to Quality – Reference to a product, service and/or transactional characteristic that significantly influences a customer in terms of quality.

CTS – Critical to Satisfaction – Expression of customers’ vital needs and can include any of the CTC, CTQ and CTD requirements.

Curvilinear Relationship – A quadratic relationship evident between two variables, that approximates a curved line on a Scatter Plot. It indicates that one variable depends on the squared value of the other.

Customer Specification Limits – Customer-defined limits for acceptable outputs. When a measured value falls outside these limits, a defect has been created.

DPM – Defects per Million – Measure of total defects per million units.

DPMO – Defects per Million Opportunities – Equals the total number of defects per unit divided by the total number of opportunities for defects per unit multiplied by 1,000,000.

DPO - Defects per Opportunity – Measure of the total defects per unit divided by the opportunities per unit.

DPU – Defects per Unit – Calculated by dividing the total number of defects by the total number of units. Also used to calculate Rolled Throughput Yield.

Degrees of Freedom – Degrees of freedom (df) of a test statistic equals the number of independent observations in a sample minus the number of population parameters that are estimated from the sample observation.

DFSS – Design for Six Sigma – Systematic methodology for incorporating VOC at product design time ensuring a Six Sigma level

DOE – Design of Experiments – Planned experiments that allow for the simultaneous statistical analysis of several PIVs to determine their effects on any measurable POV. Used to prove correlation and causation.

Deviation – The distance between a data point and the mean. Deviation measures and describes the variation in a set of data.

Discrete Data – Numeric data that is not capable of being meaningfully subdivided into more precise increments.

Discrimination – The ability of the measurement system to adequately detect the smallest tolerable changes within the process.

Distribution – A pattern or tendency depicted by randomly collected observations from a population.

DMAIC – Design, Measure, Analyze, Improve, Control – Six Sigma methodology applied to manufacturing or production processes.

Empirical Rule – A rule derived from observations that the probabilities of empirical (real world) distributions will approximate to 68%, 95% and 99.7% of the values within one, two and three standard deviations of the mean, even though the real-world distributions may not be perfectly normal.

FMEA – Failure Mode and Effects Analysis – Failure Mode and Effects Analysis is an analytical technique that allows a project team to ensure that, to the extent possible, potential failure modes and their associated causes/mechanisms have been evaluated and addressed to mitigate any risk to the customer.

Final Yield – Final Yield (YF) is a measure of the percentage of units that passed the final process test relative to the number of units that entered the process.

F-Ratio – The result of an F-Test. It indicates the measure of between-to-within variation.

Frequency – The ratio of the number of times an event occurs in a series of trials in a random experimental trial to the total number of trials in that experiment.

Frequency Distribution – A display indicating how often a particular observation or data value occurs and representing the distribution of data.

F-Test – Type of hypothesis test used to determine whether or not the within variance and between variance are the same.

Gage Control Plan – Documentation describing the strategy employed to ensure the reliability and adequacy of the measurement system in measuring input variables and monitoring output variables over the long term.

Hypothesis Test – A test in which the project team assumes an initial claim, the null hypothesis, to be true and then tests this claim against an alternative hypothesis using sample data.

I-MR Chart – A type of Control Chart for variable data that plots individual data and the moving range of the present and previous individuals.

Interaction Plot – A type of plot that graphs the averages of the output variable for each level of a factor, with the level of a second factor held constant for all combinations of levels. Interaction Plots readily show the presence of interactions; parallel lines in an Interaction Plot indicate no interaction, while greater departure from the parallel state indicates a higher degree of interaction.

KPIV – Key Process Input Variable – An input (X) that has been determined to have a statistically significant and causal relationship to the KPOV.

KPOV – Key Process Output Variable – Any outputs from a process that satisfies CTS requirements.

Linearity – The ability of the measurement system to measure over its operating range with minimal bias.

LCL – Lower Control Limit – Limit in the Control Chart that is set below the centerline, typically at -3 standard deviations, in order to determine whether or not the process is in control.

LSL – Lower Specification Limit – Limit of a tolerance range specified by a CTS. A measured value that is below this limit is considered a defect.

Mean – A measure of the central tendency of a data set. It is the sum of a set of values divided by the number of summed values. Same as average.

MSA – Measurement Systems Analysis – A series of designed tests used to assess measurement system capability.

Measurement Variability – The net effect of all the sources of measurement error that cause an observed value to deviate from the true value of the characteristic that is being measured.

Median – Center or middle number in a group of rank-ordered numbers (1, 4, 5, 9, 14)

Mode – The most frequently observed value in a dataset.

Multiple Linear Regression – A linear regression with two or more predictors. The estimation of the output variable from two or more continuous input variables using a linear relationship (straight line or plane) between the output and input variables.

Noise – The Process Input Variables (PIV’s) with settings that cannot be adjusted or controlled by the project team.

Normal Distribution – A distribution of data described by the mean and standard deviation. The curve displaying the distribution of data is shaped like a bell, with the area under the curve representing 100% of all possible observations.

Normal Probability Plot – A type of plot that represents normally distributed sample data.

Normalized Yield – Normalized Yield (YNORM) is the equalized (same) yield assigned for each process step; it is the geometric average of the Rolled Throughput Yield (RTY) for the entire process.

NP Chart – This tool tracks the number of defectives (products, parts or units that do not conform to specified standards) in each subgroup and detects the presence of special cause variation. It can only be used when the sample size is constant.

Null Hypothesis – A statement about population parameters, typically implying “no effect” or “no difference.” This statement is assumed true until sufficient evidence is presented otherwise.

OFAT – One-Factor-at-a-Time – One-Factor-at-a-Time (OFAT) is an experimental design setup, in which each factor is varied one at a time while the remaining factors are held constant. OFAT is done in order to estimate the effect of a single variable on selected fixed conditions of other variables.

Outlier – Data point that is markedly inconsistent with the rest of the data set.

Output – The result of a process or product and its measurable characteristics. Also called Process Output Variable (POV).

P Chart – A P Chart tracks the proportion of defectives (products, parts or units that do not conform to specified standards) in each subgroup and detects the presence of special cause variation. It can be used for sample sizes that are either constant or variable.

Paired Sample – A sample in which each sampling unit contains a pair of observations that are not independent of one another (e.g. sampling ages of husband and wife as a single sampling unit).

Paired t Test – A type of hypothesis test used when analyzing the difference between the means obtained from paired samples.

Pareto Chart – Type of chart that compares the frequency and/or influence of various types of problems or causes of a problem. The horizontal axis represents the different categories, and the height of the bar represents the frequency of that category. Pareto Charts prioritize possible areas for improvement and require either discrete or continuous data.

Pareto Principle – Also referred to as Pareto Effect, the Pareto Principle states that 80% of the issues are the result of 20% percent of the causes.

Pearson Correlation – The most commonly used method of computing a correlation coefficient between variables that are linearly related.

Population – The entire collection or set of objects or individuals from which a sample is drawn for analysis. Population can also mean a set of values or characteristics of the members of the population, e.g. heights of people in the world.

Pp – A capability index that measures the potential capability of a process to meet expected specification limits, or tolerance levels, in the long term, assuming the process is ideally centered (regardless of where the process is actually centered).

Ppk – A capability index that measures the potential capability of a current process to meet expected specification limits, or tolerance levels, in the long term, using the current process average.

PIV – Process Input Variable – A Process Input Variable (PIV) is a characteristic of materials, equipment, information or any other resource that is needed to carry out a process. In other words, PIV’s are the X’s and potential X’s in the Y = f(x) equation.

POV – Process Output Variable – A Process Output Variable (POV) is a characteristic of a product or service that is created by the process and is passed onto the next process step or the customer. In other words, the POV is the Y in the Y = f(x) equation.

Process Specification Limits – Limits that reflect the customer specifications as allocated to the inputs and outputs of a process. Process Specification Limits apply either to Process Input Variables (PIV’s) or to Process Output Variables (POV’s). When a PIV falls outside these limits, the process may not function as designed. When a POV falls outside these limits, the process produces a defect.

Proportion – The percentage of a data set that has a specific characteristic of interest.

Proportion of Sample – The fraction of samples which exhibit a characteristic of interest.

p-Value – The probability of rejecting the null hypothesis when it is true.

QFD – Quality Function Deployment – Quality Functional Deployment (QFD) is a systematic methodology to integrate the Voice of the Customer (VOC) into the design and delivery of goods and services.

Quartile – The four equal parts into which a rank-ordered data set can be divided.

Range – A measure of variation within a data set. It is calculated by subtracting the minimum value from the maximum value.

Regression Analysis – A statistical technique used to investigate relationships between the output variable and one or more input variables.

Repeatability – The extent to which repeated measurements, made on the same item under absolutely identical conditions, produce the same result.

Reproducibility – The extent to which repeated measurements, made on the same item under different conditions or by different people, produce the same result.

Resolution – Otherwise known as discrimination, the ability of the measurement system to adequately detect the smallest tolerable changes within the process.

Response Variable – An output variable that depends on input factors in a designed experiment. The response is the function of input variables.

Rolled Throughput Yield – Rolled Throughput Yield (YRT) is a yield metric that measures the probability that a unit of product will make it through a series of opportunities defect-free. It is calculated by multiplying the Throughput Yields at each opportunity for a defect.

Root Cause – A specific cause, usually a Process Input Variable (PIV) that has demonstrated a direct and significant influence on the Process Output Variable (POV).

Sample – A portion of the entire collection of data.

Sample Data – Observations made on items selected from a larger population.

Sample Size – The number of observations made or number of items selected from a larger population.

Sampling – Sampling is the process of selecting samples to estimate a characteristic of the population.

Scatter Plot – A type of plot used to study the relationship between two variables. It requires either continuous or discrete data. Each data point is plotted as a dot with a specific X and Y coordinate value.

Significance Level – The significance level (also called level of significance) of a hypothesis test is the maximum allowable probability of incorrectly rejecting the null hypothesis.

Simple Linear Regression – A technique in which a straight line is fitted to a set of data points to measure the effect of a single independent variable. The slope of the line is the measured impact of that variable.

SIPOC – Suppliers, Inputs, Process, Outputs and Customer. High-level process map produced as a result of project selection and definition.

Skewed – Data that is asymmetrical about the mean and is not normally distributed.

Special Cause Variation – An instance or event that impacts the process variation only under special circumstances in which the circumstance can be clearly identified, or an anomaly that is not part of the normal everyday variation encountered in the process. Sometimes referred to as unexpected variation.

Stability – Stability is the amount of variability in the bias over time. It indicates the extent to which a measurement remains constant and predictable over time with respect to accuracy and precision.

Standard Deviation – A measure of variability which describes the spread in a set of data. It is approximately the average deviation of a single data point from the mean of that data set.

Standard Normal Distribution – A special case of the normal distribution in which the mean equals zero and standard deviation equals one.

Target I-MR Chart – A type of Control Chart for variable data that plots individual data as a difference from a target and the moving range of the present and previous individual differences.

Target Proportion Value – A ratio, usually pertaining to a population, that is tested using a sample proportion.

Target Value – A value, usually a population parameter such as mean, standard deviation or proportion, that is tested using a sample.

T-Distribution – A symmetrical sampling distribution used to determine the confidence interval for means and to perform hypothesis tests on means.

Test of Association – A type of Chi-Square Hypothesis Test that examines the hypothesis of association (non-independence) between attribute variables. This procedure is used to test if the probabilities of items or subjects being classified for one variable depend upon the classification of the other variables.

Test of Independence – A type of Chi-Square Hypothesis Test that examines the hypothesis of association (non-independence) between attribute variables. This procedure is used to test if the probabilities of items or subjects being classified for one variable depend upon the classification of the other variables.

Test of One Standard Deviation Against a Constant – A type of hypothesis test for standard deviation that tests the standard deviation of a population (also known as target standard deviation) using the standard deviation of a sample.

Test of Several Proportions – A type of hypothesis test for proportions that examines whether or not two or more sample proportions of certain characteristics are independent of each other.

Test of Three or More Variances – A type of hypothesis test for standard deviation that examines the variance between three or more samples.

Test of Two Variances – A type of hypothesis test for standard deviation that examines the variance between two samples.

Test Statistic – A quantity calculated from a sample of data. The purpose of the test statistic is to obtain a p-value in a hypothesis test.

Throughput Yield – Throughput Yield (YTP) is a measure of process performance at an opportunity. It is a measure of the probability of that opportunity being correct with no rework.

Tolerance – The difference between the Upper Specification Limit (USL) and the Lower Specification Limit (LSL).

Traditional Yield – The ratio of number of units that ultimately pass through the entire process to the number of units that enter into the process. Equivalent to Final Yield, which equals output divided by input.

Treatment Combination – A specific combination of factor levels for each factor being tested in a Design of Experiments (DOE).

Trend – Gradual shift of data points in one direction.

Two-Sided Test – Hypothesis test in which the null hypothesis is rejected because the compared values are not equal to one another, regardless of whether one is smaller (or larger) than the other. For example, the alternative hypothesis for a Two-Sided Test will merely say p1 is not equal to p2, without any consideration for which is smaller (or larger).

Type I Error – The result of wrongly rejecting the null hypothesis.

Type II Error – The result of not rejecting the null hypothesis when it should have been rejected.

U Chart – A graphical display of number of defects per subgroup (part or unit) sampled with a constant or variable sample size so as to detect special causes.

UCL – See Upper Control Limit.

Uniform Distribution – Uniform distribution is a continuous probability density function, which is constant over an interval (say from “a” to “b”) and zero outside that interval. For example, while throwing a fair die, the probability of getting any number from 1 to 6 is the same (1/6) but the probability of getting a number less than 1 or more than 6 is none.

Upper Control Limit – The Upper Control Limit (UCL) is a limit in the Control Chart, set above the centerline, typically at +3 standard deviations, in order to determine if the process is in or out of control.

USL – Upper Specification Limit – The Upper Specification Limit (LSL) is the upper limit of a tolerance range specified by a customer requirement. A measured value that is above this limit is considered a defect.

Variable Data – Sometimes used in place of the term continuous data. Variable data is quantitative data that can be divided to record values at different levels of magnitude. Examples include time, distance and weight. Note: This definition is specific to Control Charts, as the data type is referred to as continuous data in other phases of the DMAIC methodology.

Variance – The spread of a set of data points, measured as the average of the squared differences between the data points and their mean. Also, the square of the standard deviation.

Variation – Used in reference to how data points are plotted with respect to a target value. Variation is commonly thought of as the measure of spread represented in a data set.

Variation Component Study – A designed experiment that allows one to partition sources of variability.

Versions – Iterations of a process map.

VOC – Voice of the Customer – Voice of the Customer (VOC) encompasses comments made by customers that describe how they feel about a company’s products, services and/or transactions. These comments typically have no associated measurable standards. The company’s responsibility is to communicate with its customers to identify these measurable standards, so that requirements can be met. These measurable standards become Customer Specification Limits.

VOP – Voice of the Process – Voice of the Process (VOP) encompasses the entire range of the output (Y) of a process when all of the X’s in the Y = f(x) equation have varied their full range.

X -Process Input Variable (PIV).

Y – Process Output Variable (POV).

Y = f(x) – The mathematical relationship that identifies the inputs (X’s) that need to be controlled and the levels they need to be set at to achieve the desired output (Y) to meet customer requirements.

Yield – The ratio of the number of units that meet certain criteria to the number of units that enter into the process. The criteria determines if a yield is Traditional Yield or another kind of yield.

YF – Final Yield.

YNORM – See Normalized Yield.

YRT – See Rolled Throughput Yield.

YTP – See Throughput Yield.

Z-Distribution – A special case of the normal distribution in which the mean equals zero and the standard deviation equals one.

Z-Score – Standardized score for measuring process performance relative to customer requirements.

Z-Table – A statistical table of probabilities associated with the Z-distribution or standard normal distribution.

Z-Transform – An equation that transfers a set of data represented by a normal distribution into a standard normal distribution (mean equals zero and standard deviation equals one) by subtracting the original mean from the data set and dividing the result by the original standard deviation.

2 Sigma – 308,537DPMO

3 Sigma – 66,807DPMO

4 Sigma – 6,210DPMO

5 Sigma – 233DPMO

6 Sigma – 3.4DPMO

101 Things Every Six Sigma Black Belt Should Know

admin — Tue, 25 Aug 2009 01:39:05 +0000

In general, a Six Sigma Black Belt should be quantitatively oriented.
With minimal guidance, the Six Sigma Black Belt should be able to use data to convert broad generalizations into actionable goals.
The Six Sigma Black Belt should be able to make the business case for attempting to accomplish these goals.
The Six Sigma Black Belt should be able to develop detailed plans for achieving these goals.
The Six Sigma Black Belt should be able to measure progress towards the goals in terms meaningful to customers and leaders.
The Six Sigma Black Belt should know how to establish control systems for maintaining the gains achieved through Six Sigma.
The Six Sigma Black Belt should understand and be able to communicate the rationale for continuous improvement, even after initial goals have been accomplished.
The Six Sigma Black Belt should be familiar with research that quantifies the benefits firms have obtained from Six Sigma.
The Six Sigma Black Belt should know or be able to find the PPM rates associated with different sigma levels (e.g., Six Sigma = 3.4 PPM)
The Six Sigma Black Belt should know the approximate relative cost of poor quality associated with various sigma levels (e.g., three sigma firms report 25% COPQ).
The Six Sigma Black Belt should understand the roles of the various people involved in change (senior leader, champion, mentor, change agent, technical leader, team leader, facilitator).
The Six Sigma Black Belt should be able to design, test, and analyze customer surveys.
The Six Sigma Black Belt should know how to quantitatively analyze data from employee and customer surveys. This includes evaluating survey reliability and validity as well as the differences between surveys.
Given two or more sets of survey data, the Six Sigma Black Belt should be able to determine if there are statistically significant differences between them.
The Six Sigma Black Belt should be able to quantify the value of customer retention.
Given a partly completed QFD matrix, the Six Sigma Black Belt should be able to complete it.
The Six Sigma Black Belt should be able to compute the value of money held or invested over
time, including present value and future value of a fixed sum.
The Six Sigma Black Belt should be able to compute present value and future value for various compounding periods.
The Six Sigma Black Belt should be able to compute the break even point for a project.
The Six Sigma Black Belt should be able to compute the net present value of cash flow streams, and to use the results to choose among competing projects.
The Six Sigma Black Belt should be able to compute the internal rate of return for cash flow streams and to use the results to choose among competing projects.
The Six Sigma Black Belt should know the COPQ rationale for Six Sigma, i.e., he should be able to explain what to do if COPQ analysis indicates that the optimum for a given process is less than Six Sigma.
The Six Sigma Black Belt should know the basic COPQ categories and be able to allocate a list of costs to the correct category.
Given a table of COPQ data over time, the Six Sigma Black Belt should be able to perform a statistical analysis of the trend.
Given a table of COPQ data over time, the Six Sigma Black Belt should be able to perform a statistical analysis of the distribution of costs among the various categories.
Given a list of tasks for a project, their times to complete, and their precedence relationships, the Six Sigma Black Belt should be able to compute the time to completion for the project, the earliest completion times, the latest completion
times and the slack times. He should also be able to identify which
tasks are on the critical path.
Give cost and time data for project tasks, the Six Sigma Black Belt should be able to compute the cost of normal and crash schedules and the minimum total cost
schedule.
The Six Sigma Black Belt should be familiar with the basic principles of benchmarking.
The Six Sigma Black Belt should be familiar with the limitations of benchmarking.
Given an organization chart and a listing of team members, process owners, and sponsors, the Six Sigma Black Belt should be able to identify projects with a low probability of success.
The Six Sigma Black Belt should be able to identify measurement scales of various metrics (nominal, ordinal, etc).
Given a metric on a particular scale, the Six Sigma Black Belt should be able to determine if a particular statistical method should be used for analysis.
Given a properly collected set of data, the Six Sigma Black Belt should be able to perform a complete measurement system analysis, including the calculation of
bias, repeatability, reproducibility, stability, discrimination (resolution)
and linearity.
Given the measurement system metrics, the Six Sigma Black Belt should know whether or not a given measurement system should be used on a given part or process.
The Six Sigma Black Belt should know the difference between computing sigma from a data set whose production sequence is known and from a data set whose production sequence is not known.
Given the results of an AIAG Gage R&R study, the Six Sigma Black Belt should be able to answer a variety of questions about the measurement system.
Given a narrative description of “as-is” and “should-be” processes, the Six
Sigma Black Belt should be able to prepare process maps.
Given a table of raw data, the Six Sigma Black Belt should be able to prepare a frequency tally sheet of the data, and to use the tally sheet data to construct a
histogram.
The Six Sigma Black Belt should be able to compute the mean and standard deviation from a grouped frequency distribution.
Given a list of problems, the Six Sigma Black Belt should be able to construct a Pareto Diagram of the problem frequencies.
Given a list which describes problems by department, the Six Sigma Black Belt should be able to construct a Crosstabulation and use the information to perform a Chi-square analysis.
Given a table of x and y data pairs, the Six Sigma Black Belt should be able to determine if the relationship is linear or non-linear.
The Six Sigma Black Belt should know how to use non-linearity’s to make products or processes more robust.
The Six Sigma Black Belt should be able to construct and interpret a run chart when given a table of data in time-ordered sequence. This includes calculating
run length, number of runs and quantitative trend evaluation.
When told the data are from an exponential or Erlang distribution the Six Sigma Black Belt should know that the run chart is preferred over the standard X control chart.
Given a set of raw data the Six Sigma Black Belt should be able to identify and compute two statistical measures each for central tendency, dispersion, and shape.
Given a set of raw data, the Six Sigma Black Belt should be able to construct a histogram.
Given a stem & leaf plot, the Six Sigma Black Belt should be able to reproduce a sample of numbers to the accuracy allowed by the plot.
Given a box plot with numbers on the key box points, the Six Sigma Black Belt should be able to identify the 25th and 75th percentile and the median.
The Six Sigma Black Belt should know when to apply enumerative statistical methods, and when not to.
The Six Sigma Black Belt should know when to apply analytic statistical methods, and when not to.
The Six Sigma Black Belt should demonstrate a grasp of basic probability concepts, such as the probability of mutually exclusive events, of dependent and independent events, of events that can occur simultaneously, etc.
The Six Sigma Black Belt should know factorials, permutations and combinations, and how to use these in commonly used probability distributions.
The Six Sigma Black Belt should be able to compute expected values for continuous and discrete random variables.
The Six Sigma Black Belt should be able to compute univariate statistics for samples.
The Six Sigma Black Belt should be able to compute confidence intervals for various statistics.
The Six Sigma Black Belt should be able to read values from a cumulative frequency ogive.
The Six Sigma Black Belt should be familiar with the commonly used probability distributions, including: hypergeometric, binomial, Poisson, normal, exponential,
chi-square, Student’s t, and F.
Given a set of data the Six Sigma Black Belt should be able to correctly identify which distribution should be used to perform a given analysis, and to use the distribution to perform the analysis.
The Six Sigma Black Belt should know that different techniques are required for analysis depending on whether a given measure (e.g., the mean) is assumed known or estimated from a sample. The Six Sigma Black Belt should choose and properly use the correct technique when provided with data and sufficient information about the data.
Given a set of subgrouped data, the Six Sigma Black Belt should be able to select and prepare the correct control charts and to determine if a given process is
in a state of statistical control.
The above should be demonstrated for data representing all of the most common control charts.
The Six Sigma Black Belt should understand the assumptions that underlie ANOVA, and be able to select and apply a transformation to the data.
The Six Sigma Black Belt should be able to identify which cause on a list of possible causes will most likely explain a non-random pattern in the regression residuals.
If shown control chart patterns, the Six Sigma Black Belt should be able to match the control chart with the correct situation (e.g., an outlier pattern vs. a gradual
trend matched to a tool breaking vs. a machine gradually warming up).
The Six Sigma Black Belt should understand the mechanics of PRE-Control.
The Six Sigma Black Belt should be able to correctly apply EWMA charts to a process with serial correlation in the data.
Given a stable set of subgrouped data, the Six Sigma Black Belt should be able to perform a complete Process Capability Analysis. This includes computing and interpreting capability indices, estimating the % failures, control limit calculations, etc.
The Six Sigma Black Belt should demonstrate an awareness of the assumptions that underlie the use of capability indices.
Given the results of a replicated 22 full-factorial experiment, the Six Sigma Black Belt should be able to complete the entire ANOVA table.
The Six Sigma Black Belt should understand the basic principles of planning a statistically designed experiment. This can be demonstrated by critiquing various
experimental plans with or without shortcomings.
Given a “clean” experimental plan, the Six Sigma Black Belt should be able to find the correct number of replicates to obtain a desired power.
The Six Sigma Black Belt should know the difference between the various types of experimental models (fixed-effects, random-effects, mixed).
The Six Sigma Black Belt should understand the concepts of randomization and blocking.
Given a set of data, the Six Sigma Black Belt should be able to perform a Latin Square analysis and interpret the results.
Ditto for one way ANOVA, two way ANOVA (with and without replicates), full and fractional factorials, and response surface designs.
Given an appropriate experimental result, the Six Sigma Black Belt should be able to compute the direction of steepest ascent.
Given a set of variables each at two levels, the Six Sigma Black Belt can determine the correct experimental layout for a screening experiment using a saturated design.
Given data for such an experiment, the Six Sigma Black Belt can identify which main effects are significant and state the effect of these factors.
Given two or more sets of responses to categorical items (e.g., customer survey responses categorized as poor, fair, good, excellent), the Six Sigma Black Belt will be able to perform a Chi-Square test to determine if the samples are
significantly different.
The Six Sigma Black Belt will understand the idea of confounding and be able to identify which two factor interactions are confounded with the significant main effects.
The Six Sigma Black Beltwill be able to state the direction of steepest ascent from experimental data.
The Six Sigma Black Belt will understand fold over designs and be able to identify the fold over design that will clear a given alias.
The Six Sigma Black Belt will know how to augment a factorial design to create a composite or central composite design.
The Six Sigma Black Belt will be able to evaluate the diagnostics for an experiment.
The Six Sigma Black Belt will be able to identify the need for a transformation in y and to apply the correct transformation.
Given a response surface equation in quadratic form, the Six Sigma Black Belt will be able to compute the stationary point.
Given data (not graphics), the Six Sigma Black Belt will be able to determine if the stationary point is a maximum, minimum or saddle point.
The Six Sigma Black Belt will be able to use a quadratic loss function to compute the cost of a given process.
The Six Sigma Black Belt will be able to conduct simple and multiple linear regression.
The Six Sigma Black Belt will be able to identify patterns in residuals from an improper regression model and to apply the correct remedy.
The Six Sigma Black Belt will understand the difference between regression and correlation studies.
The Six Sigma Black Belt will be able to perform chi-square analysis of contingency tables.
The Six Sigma Black Belt will be able to compute basic reliability statistics (mtbf, availability, etc.).
Given the failure rates for given subsystems, the Six Sigma Black Belt will be able to use reliability apportionment to set mtbf goals.
The Six Sigma Black Belt will be able to compute the reliability of series, parallel, and series-parallel system configurations.
The Six Sigma Black Belt will demonstrate the ability to create and read an FMEA analysis.
The Six Sigma Black Belt will demonstrate the ability to create and read a fault tree.
Given distributions of strength and stress, the Six Sigma Black Belt will be able to compute the probability of failure.

100. The Six Sigma Black Belt will be able to apply statistical tolerancing to set tolerances for
simple assemblies. He will know how to compare statistical tolerances
to so-called “worst case” tolerancing.

101. The Six Sigma Black Belt will be aware of the limits of the Six Sigma approach.

Normality Assumption Explained

admin — Mon, 24 Aug 2009 17:10:32 +0000

Intended Use

The normality assumption is an extremely important topic in statistics, since the vast majority of statistical tools were built theoretically upon this assumption. For example, the 1-sample and 2-sample t-tests and Z-tests, along with the corresponding confidence intervals, assume that the data were sampled from populations having normal distributions. Most linear modeling procedures, such as Regression and ANOVA, also assume that the residuals (errors) from the model are normally distributed. In addition, the most widely used control charts and process capability statistics are based upon theoretical assumptions about the normality of the process data.

Since this assumption is often a prominent part of using many statistical tools, it is often suggested that tests be run to check on the validity of this assumption. When doing so, one should always keep in mind the following points:

1. Relative importance of the normality assumption.

Most statistical tools that assume normality have additional assumptions. In the majority of cases the other assumptions are more important to the validity of the tool than the normality assumption. For example, most statistical tools are very robust to departures from normality, but it is critical that the data are collected independently.

2. The type of non-normality.

A normal distribution is symmetric, with a certain percentage of the data within 1 standard deviation, within 2 standard deviations, within 3 standard deviations, and so on. Departures from normality can mean that the data are not symmetric. However, this is not always the case. The data may be symmetric, but the percentages of data within certain bounds do not match those for a normal distribution. Both of these will fail a test for normality, but the second is much less serious than the first.

3. Data transformations.

In many cases where the data do not fit a normal distribution, transformations exist that will make the data “more normal”. The log transformation (or the Box-Cox power transformation) is very effective for skewed data. The arcsin transformation can be used for binomial proportions.

4. The effects of non-normality

In all cases, non-normality affects the probability of making a wrong decision, whether it be rejecting the null hypothesis when it is true (Type I error) or accepting the null hypothesis when it is false (Type II error). The p-value (probability of making a Type I error) associated with most statistical tools is underestimated when the assumption of normality is violated. In other words, the true p-value is somewhat larger than the reported p-value.

5. The impact of sample size.

For large samples (n >= 25), the effects of non-normality on the probabilities of making errors are minimized, due to the Central Limit Theorem. Sample size also affects the procedures you use to test for normality, which can be very erratic for small samples.

6. Normality is assumed for the population, not the sample.

When we sample data from a population, we often collect small amounts of measurements. Before we test this small set of measurements, we should consider whether we have ever collected data from this population before and, if so, include all the data in the assessment of normality.

7. Parametric tests vs non-parametric tests.

There are many non-parametric alternatives to standard statistical tests. However, it should be noted these also assume that the underlying distributions are symmetric. Most standard statistical tools that assume normality will also work fine if the data are symmetric. Also, consider that the power of the non-parametric tests only approaches the power of the standard parametric tests when the sample size is very large. So, given the choice between the two, if the data are fairly symmetric, the standard parametric tests are better choices than the non-parametric alternatives.

What is the bottom line? If you use a statistical tool that assumes normality, you can and probably should test this assumption. Use a normal probability plot, a statistical test, AND a histogram. Never use a statistical test for normality without also using either a histogram (best choice) or a probability plot. If the test for normality fails, check for symmetry with the histogram or probability plot. If you have a large sample, the data would have to be extremely skewed before there is cause for concern. For skewed data, try a log (or power) transformation. With small samples, you are taking chances with any tool, including the probability plot, the test for normality and the histogram. Also, non-parametric alternatives have little discriminatory power with small samples. If you use a statistical tool that assumes normality, and the test fails, remember that the p-value you see will be smaller than it actually should be. This is only cause for concern when the p-value is marginally significant. You might want to run the non-parametric alternative (if there is one) and see if the results agree. And, consider the practical significance as well as the statistical significance.

How it Fits With the Breakthrough Strategy (DMAIC)

The Normality Assumption is assumed for various tools that are utilized in the following phases of the Breakthrough Strategy,

Measure Phase

Analyze Phase

Improve Phase

Control Phase

Measure Phase

In the Measure Phase of a Black Belt project, the Black Belt must perform an initial capability study to obtain a baseline for the process. This capability study should incorporate control charts to verify that the process was stable, as well as capability indices, which gage its performance. Both the control charts and the capability statistics are theoretically founded upon an assumption of normality.

As for control charts, the effects of non-normality show up in the probabilities associated with falsely failing the tests for special causes. It has been shown (Wheeler) that these effects are minimal for Xbar charts, even for very small (n < 5) subgroup sizes, provided the data are not extremely skewed. Even in cases of extremely skewed data, the effects are minimal if the subgroup size is 5 or more.

Process capability indices themselves do not assume normality – they are simply ratios of distances. However, if one makes inferences about the defect rate based upon these indices, these inferences can be in error if the data are extremely skewed. One fortunate aspect of process capability studies is that it is standard practice to base them upon at least 100 observations, therefore lessening the impact of the normality assumption.

Minitab’s capability study shows a histogram of the sample data, along with capability indices, which are used to estimate process performance. The histogram has a normal curve superimposed over it to make it easier to assess the normality of the data. Minitab also has a nice facility for finding the optimum power transformation for use with data in subgroups. For cases with heavily skewed data, Minitab will also calculate process capability assuming a Weibull distribution instead of a normal distribution.

Analyze Phase

In the Analyze Phase of a Black Belt project, the Black Belt must isolate variables which exert leverage on the CTQ. These leverage variables are uncovered through the use of various statistical tools designed to detect differences in means, differences in variances, patterns in means, or patterns in variances, in the case where the CTQ is continuous.

Most of these tools assume normality. For many, there are non-parametric alternatives. As mentioned earlier, one should use good sense and judgement when deciding which test is more appropriate. It is a good idea to test for normality. If the test fails, check the symmetry of the data. With a large (n >= 25) sample, the parametric tools are better choices than the non-parametric alternatives, unless the data are excessively skewed.

For example, the 1-sample and 2-sample t-test (and Z-test) also assume that the samples are independent, that the measurement scale is continuous, and that the variances (for the 2-sample case) are equal. Of this set of assumptions, the least important is the assumption of normality, since these tests are very robust to the assumption of normality, provided the samples are large and that the underlying distribution(s) is (are) fairly symmetric. It is interesting to note that the non-parametric alternatives to the t and Z tests also assume that the underlying distributions are symmetric.

Similarly, linear modeling procedures such as ANOVA and Regression also assume that the residuals (errors) are independent, identically distributed, with a continuous measurement scale. Of these assumptions, the least important is again the assumption of normality. Both ANOVA and Regression are very robust to this assumption, provided the underlying distribution of the residuals is fairly symmetric.

When using these tools, don’t forget to try data transformations, especially when the data are skewed. Keep in mind that the p-values you see may be underestimated if the data are not from a normal distribution. The effect on the p-values is minimized for large samples. If a standard parametric test is used and the reported p-value is marginally significant, then the actual p-value may be marginally insignificant. When using ANY statistical tool, one should ALWAYS consider the practical significance of the result as well as the statistical significance of the result before passing final judgement.

Improve Phase

In the Improve Phase, the Black Belt will often use designed experiments to make dramatic improvements in the performance of the CTQ. A designed experiment is a procedure for simultaneously altering all of the leverage variables discovered in the Analyze Phase and observing what effects these changes have on the CTQ. The Black Belt must determine exactly which leverage variables are critical to improving the performance of the CTQ, and establish settings for those critical variables.

In order to determine whether an effect from a leverage variable, or an interaction between 2 or more leverage variables, is statistically significant, the Black Belt will often utilize an ANOVA table, or a Pareto chart of effects, or a normal plot of effects. The ANOVA table displays p-values for assessing the statistical significance of model components. The Pareto chart of effects has a cutoff line to show which effects are statistically significant. The normal plot of effects is another tool for judging which effects are statistically significant.

All of these methods assume that the residuals after fitting the model are from a normal distribution. Keep in mind that these methods are pretty robust to non-normal data, but it would still be wise to check a histogram of the residuals to be sure there are no extreme departures from normality, or more importantly, are not excessively skewed. If the data are heavily skewed, use a transformation. Also, bear in mind that the p-values in the ANOVA table may be slightly underestimated, and that the cutoff line in the Pareto chart is slightly higher than it should be. In other words, some effects that are observed to be marginally statistically significant could actually be marginally insignificant. In any case where an effect is marginally significant, or marginally insignificant, from a statistical point of view, one should always ask whether it is of practical significance before passing final judgement.

Control Phase

In the Control Phase, the Black Belt devises a control plan for each of the critical leverage variables from the Improve Phase. These are the variables which truly drive the relationship Y = f(X1,X2,…,Xn). In order to maintain the gains in performance for the CTQ, or Y, the X variables must be controlled. Part of the control plan is a sampling scheme for each X, coupled with a control chart to detect whether each X is staying on target.

Control charts for continuous data assume the data are from a normal distribution, although control charts have been shown to be very robust to the assumption of normality, in particular the Xbar chart. A simulation study found in Wheeler shows that even for subgroups of size 3, the Xbar chart is robust to non-normality except for excessively skewed data. For subgroups of size 5, the Xbar chart is robust even if the underlying data are excessively skewed.

If the I chart (subgroup size = 1) is used, the effects of non-normality will be seen in an elevated rate of false alarms from the tests for special causes. These tests are designed to have low probabilities of false alarms, based upon a normal distribution. Depending upon the type of departure from normality, certain tests will exhibit higher false alarm rates.

As with the tools mentioned earlier, you can transform the data, which is particularly effective for skewed data. Minitab has a facility for selecting the optimum power transformation to use with data in subgroups. Use this feature for skewed data.

Considerations for the Normality Assumption

The use of statistical tools does not follow some exact cookbook. By their very nature, there is always an element of error associated with these tools. The same is true about the assumption of normality. No one can make the claim that they have ever collected data from an exact normal distribution, because it does not exist. However, many naturally occurring phenomena follow a very close approximate normal distribution. So, when assessing the validity of the normality assumption, keep in mind the points outlined earlier:

1. Relative importance of the normality assumption.

2. The type of non-normality.

3. Data transformations.

4. The effects of non-normality.

5. The impact of sample size.
6. Normality is assumed for the population, not the sample.
7. Parametric tests vs non-parametric tests.

Cookbook for Checking the Normality Assumption

Minitab has several tools for assessing normality. The statistical tests for normality should never be used by themselves – they should always be accompanied by either a probability plot or a histogram.

To check normality for a sample of data

Go to the Stat menu, select Basic Statistics.
Select Display Descriptive Statistics
In the Variables box, enter the columns you want to check for normality. Note, each column you enter will generate its own display.
Click the Graphs button.
Check the box for Graphical Summary, then click Okay.
You will see a display as shown below. Check the p-value associated with the Anderson-Darling test for normality. Also check the histogram for symmetry in the data.

Equal Variance Assumption Explained

admin — Sun, 23 Aug 2009 01:55:24 +0000

The equal variance assumption is important in statistics because it applies to two of the most widely used tools, the two-sample t-test, and Analysis of Variance (ANOVA).
Since this assumption plays a prominent role when utilizing these popular tools, it is often suggested that tests be run to check on the validity of this assumption. When doing so, one should always keep in mind the following points:

1. Robustness when samples sizes are equal.

Both the two-sample t-test and ANOVA are very robust to the equal variance assumption when the sample sizes are equal, or nearly equal.

2. Alternative procedures.

The two-sample t-test can be used either with or without the assumption of equal variances. In fact, the default method in Minitab does not assume equal variances.

3. Data transformations.

If the variances of the samples are correlated with the size of the data (as Y increases, the variance of Y increases), it may be possible to use a log transformation (or the Box-Cox power transformation) to correct the problem.

4. The effects of unequal variances.

In all cases, unequal variances affect the overall estimate of the error variance. This in turn affects the corresponding t or F statistics, which in turn affects the reported p-values. The p-value (probability of making a Type I error) is underestimated when the assumption of equal variances is violated. In other words, the true p-value is somewhat larger than the reported p-value.

5. Parametric tests vs non-parametric tests.

There are non-parametric alternatives to both the two-sample t-test and ANOVA. However, it should be noted these also assume that the underlying distributions are symmetric, with the same shape. Stating that the distributions have the same shape is tantamount to stating that they have the same variance. Since both the parametric and the non-parametric tests assume basically the same thing, the parametric tests are preferred, since they are uniformly more powerful.

6. Differences in variance are also desirable.

One of the major goals of every project should be to reduce the variation in the CTQ, or Y. Don’t forget that the two-sample t-test and ANOVA are both methods for detecting changes in the mean of Y. If changes in the variation are observed, these are important, perhaps more important than the changes in the mean one is testing for. These differences should be studied to determine if they are consistent.

What is the bottom line? If you use a statistical tool that assumes equal variance, you can and probably should test this assumption. Remember that if the sample sizes are equal, or nearly equal, this assumption can be relaxed a great deal. Also, as a rule of thumb, even when the sample sizes are not nearly equal, there is usually no problem provided the largest sample standard deviation is not more than twice the size of the smallest sample standard deviation. When there is a problem, remember that what may be a “problem” as far as testing for differences in the mean of Y can also be a “solution” for determining ways to reduce the variation in Y. Be sure to check whether the variances increase as the size of the data increases. This is not at all uncommon, and can be remedied easily with a log or Box-Cox power transformation of the data. If all else fails, remember that the p-value you see will be smaller than it actually should be. This is only cause for concern when the p-value is marginally significant. You might want to run the non-parametric alternative (if there is one) and see if the results agree. And, consider the practical significance as well as the statistical significance.

How it Fits with the Breakthrough Strategy (DMAIC)

The Normality Assumption is assumed for tools that are utilized in the following phases of the Breakthrough Strategy,

Analyze Phase

Two of the prominent tools for detecting differences in means are the two-sample t-test and ANOVA. The assumptions for both of these are the same, since the two-sample t-test is just a special case of one-way ANOVA. Both of these tools also have non-parametric alternatives, at least in some cases. The t-test also has an alternative where equal variances are not assumed. There are non-parametric alternatives for one-way and two-way ANOVA. As mentioned earlier, one should use good sense and judgement when deciding which test is more appropriate.

It is a good idea to test for equal variances. If the test fails, check the ratio of the largest sample standard deviation to the smallest sample standard deviation. If the sample sizes are equal, or nearly equal, there should not be a problem unless this ratio is larger than 4. Even when the sample sizes are not nearly equal, there should not be a problem unless this ratio is greater than 2. And, more importantly, remember that you are using these tools to detect changes in the mean of Y, and the “problem” you are having is that you have detected changes in the variation of Y at the same time. Of course, that may help you satisfy the ultimate goal of reducing the variation of Y.

When using these tools, don’t forget to try data transformations, especially when the sample variances increase as the sample means increase. Keep in mind that the p-values you see may be underestimated if the variances are not equal. If a t-test or ANOVA are used and the reported p-value is marginally significant, then the actual p-value may be marginally insignificant. When using ANY statistical tool, one should ALWAYS consider the practical significance of the result as well as the statistical significance of the result before passing final judgement.

Improve Phase

In order to determine whether an effect from a leverage variable, or an interaction between 2 or more leverage variables, is statistically significant, the Black Belt will often utilize an ANOVA table, or a Pareto chart of effects, or a normal plot of effects. The results from all of these are based upon the estimate of the error variance, whicother words, the sample sizes are all equal. Thus, the equal variances assumption can be relaxed for balanced experiments.

Still, it is good practice to check for radical departures from the equal variance assumption. In order to do so you must remember one thing – it is the variances of the residuals that are assumed to be equal, not the variances of the Y values. This means that you would need to organize the residuals into groups based upon the values of the factors, one group for each term in the model, and then get an estimate of the variance from each group. For a one-way ANOVA this is not too hard to do. For a two-way ANOVA, it is still not difficult. But, for more than two factors, this becomes increasingly difficult to do. An alternative would be to examine the residuals for extreme outliers, or plot the residuals against each factor as well against the fits. These plots will at least show whether any of the factors have main effects on the variability of the residuals, as well as determining whether the residual variance is correlated with the response.

Also, bear in mind that the p-values in the ANOVA table may be slightly underestimated, and that the cutoff line in the Pareto chart is slightly higher than it should be. In other words, some effects that are observed to be marginally statistically significant could actually be marginally insignificant. In any case where an effect is marginally significant, or marginally insignificant, from a statistical point of view, one should always ask whether it is of practical significance before passing final judgement.

Considerations for the Equal Variance Assumption

The use of statistical tools does not follow some exact cookbook. By their very nature, there is always an element of error associated with these tools. The same is true about the assumption of equal variances. Since the Y variable for both the two-sample t-test and ANOVA is the same for all samples, it is not likely that the variances will differ greatly from one sample to another. On the other hand, since one of the goals of a good project is to reduce the variation in Y, extreme differences in variability should be studied to determine why they occurred. It is not uncommon to observe that the variance of Y increases as the mean of Y increases, a condition that can be easily remedied with a data transformation. So, when assessing the validity of the equal variances assumption, keep in mind the points outlined earlier:

1. Robustness when samples sizes are equal.

2. Alternative procedures.

3. Data transformations.

4. The effects of unequal variances.

5. Parametric tests vs non-parametric tests.

6. Changes in variation can be desirable.

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